Answer:
![f'(2)=\lim_{h \rightarrow 0} \frac{-1}{4(4+h)}](https://tex.z-dn.net/?f=f%27%282%29%3D%5Clim_%7Bh%20%5Crightarrow%200%7D%20%5Cfrac%7B-1%7D%7B4%284%2Bh%29%7D)
Step-by-step explanation:
The definition of derivative is:
![f'(x)=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Clim_%7Bh%20%5Crightarrow%200%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D)
So it asks us not to evaluate the limit part but to simplify the fraction part.
So let's focus on just:
![\frac{f(x+h)-f(x)}{h}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D)
We are given
.
So
(I just replaced the
's with
's.)
Now since we want to find it a
. I'm going to replace my x's with 2:
So instead we will look at:
![\frac{f(2+h)-f(2)}{h}](https://tex.z-dn.net/?f=%5Cfrac%7Bf%282%2Bh%29-f%282%29%7D%7Bh%7D)
![\frac{\frac{1}{(2+h)+2}-\frac{1}{2+2}}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cfrac%7B1%7D%7B%282%2Bh%29%2B2%7D-%5Cfrac%7B1%7D%7B2%2B2%7D%7D%7Bh%7D)
Let's simplify some of the addition that we can in the denominators of the mini-fractions:
![\frac{\frac{1}{4+h}-\frac{1}{4}}{h}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cfrac%7B1%7D%7B4%2Bh%7D-%5Cfrac%7B1%7D%7B4%7D%7D%7Bh%7D)
Now division by
can be written as multiplication by
:
![\frac{1}{h}(\frac{1}{4+h}-\frac{1}{4})](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bh%7D%28%5Cfrac%7B1%7D%7B4%2Bh%7D-%5Cfrac%7B1%7D%7B4%7D%29)
Let's combine the fractions inside the ( ).
I will multiply the first fraction by
.
I will multiply the second fraction by
.
We are going to do this so we have the same denominator:
![\frac{1}{h}(\frac{4}{4(4+h)}-\frac{4+h}{4(4+h)})](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bh%7D%28%5Cfrac%7B4%7D%7B4%284%2Bh%29%7D-%5Cfrac%7B4%2Bh%7D%7B4%284%2Bh%29%7D%29)
Now we have the same denominator inside the ( ) and can combine those fractions:
![\frac{1}{h}(\frac{4-(h+4)}{4(4+h)})](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bh%7D%28%5Cfrac%7B4-%28h%2B4%29%7D%7B4%284%2Bh%29%7D%29)
Let's simplify the numerator in the ( ).
![\frac{1}{h}(\frac{-h}{4(4+h)})](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bh%7D%28%5Cfrac%7B-h%7D%7B4%284%2Bh%29%7D%29)
Now you should see a common factor to cancel. That is we have that
. So we can write that:
![\frac{1}{h}(\frac{-h}{4(4+h)})=\frac{-1}{4(4+h)}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bh%7D%28%5Cfrac%7B-h%7D%7B4%284%2Bh%29%7D%29%3D%5Cfrac%7B-1%7D%7B4%284%2Bh%29%7D)
So the answer we are looking for is:
![f'(2)=\lim_{h \rightarrow 0} \frac{-1}{4(4+h)}](https://tex.z-dn.net/?f=f%27%282%29%3D%5Clim_%7Bh%20%5Crightarrow%200%7D%20%5Cfrac%7B-1%7D%7B4%284%2Bh%29%7D)